 Can't tell us about dark matter or other fundamental fields that may not even be a significant fraction of dark matter But before we start let me finish what I was saying About the the puzzle that we have okay, it's still a puzzle. I have no idea what's going on here but if you remember we basically showed that if you have an Object, which is extremely compact then the intermediate signal must be dominated by By the photosphere so the photosphere modes which are the black hole quasi normal modes, okay, so if you throw something into a Very compact object the signal would look something like this and It has to be exactly the same as a signal coming from a black hole, okay We are not allowed to tell the difference Up until times when the when the gravitation wave has had time to reach the surface of the object, okay So from this critical point lying somewhere there onwards Then the object may respond and we might see imprints of the surface, okay Now do the the puzzle is the following If you change the boundary conditions of your problem like for instance Instead of a black hole now I have a star The star is some surface and you are imposing boundary conditions Not at the horizon, but either at the surface or regularity at the origin so they change completely Okay, and because the boundary conditions change completely the quasi normal frequencies of these all of this object by the way this object has a surface at the horizon radius Plus an epsilon, okay, so it's not a black hole very close to being a black hole boundary conditions completely different mode structure completely different Okay, you can check that with a notebook. I sent you once again As an example Let me remind you that for a black hole VH These numbers here look something like 0.374 minus 0.0899, okay, this is the lowest mode For a star whose surface lies at with an epsilon say 10 to minus 6 these numbers change into I'm making up, okay making something up 0.2 minus 10 to minus 6 I Actually, this is a number that came up in simulations so the paradox is if I If I look at the sine wave exponentially dumb sine wave these numbers match These frequencies both for the black hole and for the compact object, right? So what's happening? we're in the spectral information of my PDs is This or is this actually you see if I have a compact object this number is not there In the mode spectrum, but it is in the time domain waveform. It's this one, right? So what's going on? Half of the of the answer we know it So let me show you the simple delta function that we solve for so this We had this delta function potential and we evolved Initial with initial data at x not, okay, and if you remember the the The frequency was minus V naught over 2i, okay So if you evolve this this problem in time you get something like this, okay? so you get So this initial condition is some Gaussian This is not of kind of funny, okay So you see all of the so these are the first two echoes first signal. I'm sitting somewhere to your right Okay, very far away first signal Then an echo second echo and here I have like a hundred echoes, okay? And what you see is the signal itself Decays so this is fit to this value which is so V naught was one-half So omega should be one-fourth minus one-fourth of i, okay? It matches very nicely the way the poles the individual poles decays with time Okay Good. So this is the the analog of the black hole Because you know, okay, but then you see at late times It sets in To a new behavior. So this number here 0.144 minus 0 blah blah is the quasi normal mode of the composite system because you see now I put a barrier here without the barrier. It's a black hole system We did everything we had to do now. I'm placing a barrier here and I'm going to get echoes, right? So the very late time behavior Sorry, this guy is governed by the quasi normal modes of the of the full system. This is fine And it agrees with this the problem is well the puzzle the puzzle is Where exactly is this information? I? Have absolutely no idea, okay? So I have nothing else to add to this problem, but it's an interesting open there must be a tool as Tool in spectral analysis that tells you well this you had to find this, okay? And actually sorry, I was going to forget You actually this problem this echo business didn't start with trying to find alternatives to black holes It started by trying to understand how buckles in the galaxy respond Okay, and yesterday I was telling you that we did this experiment where we placed a black hole inside a shell And as the shell was made larger and larger the modes of the system Became more and more different from that of an isolated buckle Exactly the same thing happens if you do a time domain waveform you see something like this So the the the modes which become so different are actually only the late time behavior of the signal The early time response is exactly the same as that of a single buckle with nothing around it, right? so if you place a galaxy and Too many stars around the black hole you're going to see echoes caused by the Reflection of the ring down waves generated at the photos here and interacting with the star with the stars gravitational field, okay? Very good But today I'd like to discuss well now I'd like to discuss a new phenomena or another phenomena connected to black holes That instead of telling you about the structure the near horizon structure of of these guys Is going to tell you About the not so near horizon structure, but about the ergo region so negative energy states This phenomenon is known as super radiance. It's not new or unique to black holes Okay, actually we knew about super radiance decades ago. The first example. I know of Is this it was done? It was it's an example taken from fluid dynamics. It was worked out in 57 and Landau and wish it's have half a page on this which means two days of work So the example is the following you take two fluids Okay, this lower half plane Suppose it's water and the upper half plane this guy here is air Okay, like ocean and air and the air is moving with respect to the ocean, okay? Now a fish in the ocean is doing an experiment The fish is going to stand a sound wave towards the interface Okay, and then he's going to record how much of that sound of how much of the amplitude of that wave is reflected back, okay Now if you do the calculation, which is not that tough, but I mean it does require attention to details You will find well the fish will find that the amplitude of the reflected wave So no change in frequency The amplitude of the reflected wave is larger than the amplitude of the way we sent in When the relative velocity is larger than the local speed of sound Okay, so if this interface is moving supersonically The sound wave is extracting energy away from the motion and that's of course how the fish gets energy back This is a kind of shedding of effect if you wish and in fact you could you know You could place a bunch of phenomena under this super avian's category I'm not going to be discussing Translational this is a translational kind of thing, okay? I'm going to discuss spinning black also for that. I need another related phenomena, which is rotational super avian's and There's one example that we are very familiar with that looks like rotational super avian's and that's the earth-moon system Okay, there's really no wave involved But if you pay attention to the to the what's happening you'll realize immediately that it has to be very generic now in the earth-moon system Why you could it could be the earth Sun, but let me focus on the earth moon Okay, the moon so this explanation by the way was given by the Sun of Darwin So the first explanation for tides So you have the earth it's been it spins around its axis once every 24 hours Okay, the moon is going around the earth once every roughly 28 days So during one day basically the moon is standing still Okay So I'll admit that now the earth has oceans Around it and the moon is basically pulling tides on it, right? And you see the tides look something like this. I'm of course exaggerating everything and so on But because everything would be nice and smooth if there was nothing no friction or nothing The the tide earth axis would basically point towards the earth moon axis, right? It would be a lot everything would be aligned big but because there's friction between the crust and the oceans There's a very small angle Darwin called it the title angle phi Between earth moon and earth tides. Okay, very small 10 to minus 6 whatever Okay, so what is this doing exactly? Well First because there's friction The day has to be getting longer Okay, right as years go by this is kind of trivial what's not so trivial is Well, but if the day is getting longer the earth is slowing down of course But where the heck is the angular momentum going? It's an isolated system has to conserve angular momentum Well, the angular momentum is going to the moon So there's a transfer of angular momentum actually caused by this lump of ocean That causes the moon to gain angular momentum if you do the calculation. It's half a line It means that the moon is getting farther away from the earth. This has been measured. Okay People put mirrors on the moon. They throw lasers in and they measure it's two centimeters a year or so. It's getting farther When will the process stop it has to stop someday By the way, does anybody know when will the process stop? It's my last lecture. I need to venture Well, it started because It started because the earth is spinning very fast, right? So it will stop when the day of the earth is equal to The rotation period of the moon, right? Then there's no tidal angles. There's nothing everything is aligned So if the earth has always the same face towards the moon That's it and now you can think oh wait a minute is this why the moon has the same face towards the earth It is so this process was active on the moon millions of years ago. Actually, there's a nice book Called the cosmic stories or by Calvino on this issue. So many millions of years back The moon was much closer to the earth people could go So sorry, yeah, but anyway So now you can you kind of understand that this should happen For any spinning object. You see this is really just friction. The universe doesn't really like relative motion, right? So everything needs to be synchronized somehow So the question is does this happen for? Blackhole space times or any other space I'm or not Actually, does it happen if I place a cylinder here and I spin it does this happen? And the answer is yes, it does if you put a cylinder rotating here And you throw sound waves at the cylinder as long as there's a coupling between the cylinder and the sound wave Superavius must occur. This was measured last year It's it was not a cylinder It was what people called an acoustic dumb hole It's basically a version of a a black hole in water, okay? So let me show you that it has to happen Even if you have a black hole, so I'm repeating the same story. I'm taking a very simple Field content just a massive scalar field You remember we basically separated and decoupled this equation in partial It was very simple because there's spherical symmetry, so you could expand in spherical harmonics You do exactly the same thing here and you will find well the background is not spherically symmetric So you need to be a bit more refined, but separation of variables work So if you try these sunsets if you try Phi equals some radial function times some angular function and oops An harmonic time dependence So theta and Phi are the angular variables of theta as you move for Phi Okay, then you will find the following you can separate variables You get two Couple equations one for the radial wave function are another for the angular function Capital Theta they look something like this So I'm doing this in Kerr Kerr space. I'm okay Delta d dr of r Plus k squared over Delta. I'll tell you what these numbers are Minus a tilde minus a was the spin of the buckle Omega is this frequency of the wave plus 2m am Omega R equals zero so you see This is fine. I really got a decoupled Equation for our separated and for the angular part one over sine of theta Did the theta of sine of theta? Did the theta of this capital theta? plus a Squared Omega squared cosine square of theta minus m squared over cosine square of theta. Sorry sine squared Plus a capital theta equals zero Okay, and is he as he moves all number this guy here Omega is this frequency This quantity k is r squared plus a squared Omega Minus a m a tilde is a separation constant and it's a plus the mass R squared and Delta is in the metric the background metric. Okay, it defines the horizon It's this guy Okay So everything now is in terms of spin and black or mass and you will notice that the only thing you don't know is the separation constant a Okay, right Of course you do so by the way if Omega is zero This really is just a legend Equation so solutions are spherical harmonics. Okay When a is not zero this is called a spin weighted spherical harmonic actually a spin zero spherical harmonic spheroidal harmonic, sorry In the general case, of course, you find the separation constant just requiring regularity of the angular function At zero and pi. Okay. Yeah, sorry. Sorry spin zero spheroidal harmonics. It's not the same because of this term Okay, they appear in a lot of Things in physics actually well from now has The s zero I mean there you you get similar stuff for s one and s two well from now coded this equation for s zero So there's a you can call spheroidal harmonic for spin zero fields Very good So this is obviously now if Omega is real if you're just throwing stuff in a Standard certainly reveal problem. You can solve it using using the usual tools. Let me look at the radial equation So if you try to understand the behavior at the horizon the horizon by the way our roots of this delta Okay, you do a standard local analysis for a b for a b news analysis. So you search for a parameter beta okay, and You find that beta is plus or minus i r plus squared plus a squared Omega minus a m over r plus minus r minus This when r goes to r plus Okay At infinity your wave function our behaves as 1 over r A in so it's an in going component plus an outgoing component the usual stuff So the only difference with respect to non spinning buckles really is the horizon Dependence and you see that now Because of the spin. It's not really a standard in going wave with Kind of an easy dependence Very good. So we can try to repeat exactly what we did for schwarzschild which is using the wrong scan of two independent solutions and see if there's energy conservation, right Now this time you see there's a term that has first derivatives So we can't really say all the runs can is constant, but we know it's Abel's formula that tells us that the runs can Omega so for a function of This type P y prime plus Q y Equal zero then the runs can between two solutions is some constant times this right Since P is not zero. Okay, it's not a constant, but I know how it changes. So I'm going to take the solution The complex conjugate of this guy so psi one goes like this r minus r plus minus i gamma and 1 over r In e to the minus i omega r plus a out e to the plus i omega r Where gamma is this guy R plus squared plus a squared over r plus minus r minus Omega minus m capital omega you'll find the following you will find that The runs can times the delta function this guy has to be a constant. Let me call it k naught This comes from Abel's formula this one, okay Now you can evaluate the runs can close to the horizon close to infinity and Match them using this behavior and you'll find the following at the horizon The runs can is this 2i gamma over r minus r plus at infinity the runs can is 2i omega over r squared a in squared minus a out squared if you match Using this formula, then you'll find the following a in minus a out Has to be equal to r plus squared plus a squared over omega omega minus m capital omega Okay, so this is the final result we wanted now if you look at the what exactly in and they out are Well, they're just the in going piece the thing that you threw in Okay, when you were far away from the buckle and the thing that comes out And that you measure when you're far away from the buckle Okay So what this tells you is that if you're throwing sufficiently low frequency waves This is a wave a scalar wave. You can do it for vectors and what what not okay if omega Is small omega is smaller than the angle of frequent than the angle of velocity of your buckle then this guy is negative and This means that a in has to be smaller than a out. Okay, so super radians If omega smaller than m omega, but if you think about it this the small omega really is the analog of The angle of velocity of the moon, right? So this is a kind of a nice Way to think about this you're sending in a wave which is rotating. I mean omega is basically the angle of velocity of the wave So it's rotating slower Then then then the buckle right so it's extracting this rotational energy away from the buckle Which is which would be the earth in the in the slide, right? This doesn't tell you how much a in and a out are you can compute it Numerically, it's in the notebook again and the values look like this. Okay So So for scalar waves This curve here what the black curve are numerical results. Okay Dash lines are some low frequency approximation to the to the numerics The maximum amplification factor so Z is really not very encouraging for scalar waves We you get around zero point four percent Extra, it's not that much right For a vector you get up to four percent if you fine-tune the frequency of the way that you send in For tensors the thing is amazing if you fine-tune the frequency of the of your gravitational wave You get back a hundred and forty seven percent More than you sent in so it's a generous energy extraction Okay, this was all done for massless fields, okay So you might ask okay, that's fine. I send in a wave the way of God's gets amplified. It's funny I mean, it's not really that important, right? You don't need to tune it, but I'm just saying the amplification factors So this is always this always gives you super radians if Omega is small But the question is how big is super radians? But so if you look at the here for instance the gravitational case, okay, if Omega so this has a black hole spinning at close to the maximum possible value, right? If the frequency is too low Z is still positive. So they're still super radians, but now you're talking about ten to minus three percent So that's nothing if you fine-tune up to this point then we're talking about something interesting, right? Okay, but so the interesting thing is that this has huge applications. Well, or at least very very interesting applications. I Will discuss dark matter physics, but there's others for example actually the root of all these applications is not connected to super radiant itself or only but You need to add an extra ingredient the ingredient is very simple If you have the system it's spinning it amplifies waves now if you try to enclose the system So you have a spinning buckle Okay Now if you trap the system in a cavity What is going to happen? Well? You throw in a wave right You expect this wave to be amplified so you get something more out But then it's trapped right so it has to fall back again. It reflects here Balls back in gets amplified and so on and so forth. So this is again an exponential cascade and this is known as the black hole bulb There are space times like ADS that actually give you this mirror for free So any spinning buckle in ADS or at least Most of them should be unstable against this mechanism. So it's an interesting thing in ADS That's a good question in flat space. We know I think in ADS so So the angle momentum should be transferred to a cloud. So if this is a gravitational process Any small fluctuation of order epsilon here It's going to extract the angle of momentum until the process stops the process stops when this condition is verified, right? So at the end what you should be getting is some sort of cloud of Gravitons outside the black hole geometry Right so you extracted the angle of momentum and it was deposited in a moon outside In a way that the moon and the black hole always have the same face, right there. They are co-rotating Having said this, it's I don't think we know for sure if the process is fully stationary in ADS There are reasons to suspect that this cloud is actually Also unstable at the nonlinear level. There are there was a simulation a few months ago That seems to see this so there is no truly stationary state for this instability It's just going to break up in smaller and smaller pieces But I don't think we have a definite answer in the fact case. It's somewhat simpler and I'm going to discuss that Okay So I started I think the discussion two days ago by saying that one of the reasons why we're interested in black hole physics is because well One of the big open problems, which is dark matter We only have access to it through the gravitational channel So it kind of makes sense that we try to understand first gravity a lot better than than we do right now, right? There's some solutions that don't require a super radiance for that for example maybe dark matter all Is all under the form of buckles? Many black holes primordial black holes that were generated in the universe. They are hard to see nowadays. It's one possibility Another possibility that does also doesn't use super radiance uses the in spiral if you have dark matter in the universe And now you're merging two black holes or two neutral stars, right? They are merging in an environment, which is not empty So two things are going to happen first these waggles accrete the dark matter that they find in their path, right? They are following dark matter. They are greeting But when they do this they're also dragging behind all of the dark matter that they see right? They are pulling they need to pull all of that matter. They see they this is called gravitational drag So there's two effects and the two effects basically Contributing the same way they make the spiral proceed faster Right? So one way you have to check if dark matter is present in kind of significant quantities is by Trying to match the way the spiral proceeds with that of a black hole in vacuum if it doesn't fit well Maybe some non-trivial density of dark matter will do the job. It's different from friend. Yeah friend Dragging just means that things have to follow what the black hole is doing gravitational drag Gravitational drag Well, it's gravitational drag So if if you think about my gravitational field if I'm standing still it attracts all the molecules in this room in equal way Right, but if I'm moving right now you're going to think well these guys behind me are pulled So these guys behind me I'm going to create a wake in the density of guys behind me the guys in front Of course still don't know that I'm moving the guys behind me do you create the density? Disturbance and that density disturbance Produces its own gravitational field and that's pulling me back. So it's a non Spherical is symmetric disturbance Okay, it's called gravitational drag. It's really just drag very good Okay, but I'm going to discuss super aliens and for a reason. Okay, the reason is that many dark matter candidates Interact very very weakly with the standard model fields. Okay Very weakly indeed. So this is a plot. Well, this is an exclusion plot. Don't ask me about the experiment I have no idea what they are at least most of them. Okay, but this shows what happens if dark matter comes under the form of a Scalar actually an action that couples to the Maxwell field So this is the coupling strength as a function of the scale the mass of the scalar Okay, and the units here are 10 to the minus 5 electron volt here then to minus 15 Okay, and here the same thing for a vector field some hidden Vector sector couples to the Maxwell field again. This is the strength of the coupling I mean if there's no coupling, there's no way to see it, right? At least non-gravitational. So there is a coupling and As a function of the mass and these are exclusion plots and there's one thing that you notice immediately We can only using Standard model physics we you can only constrain stuff that you see right so at zero coupling It's impossible to constrain whatever right Okay, so I'm going to try to convince you now that black holes are the ideal tool To see things that are impossible to see otherwise, right? So even if the coupling to standard model is zero The equivalence principle of GR tells you that even dark matter falls in the same way as everything else And we're going to use this property to try to see the unseen. Okay So basically, I'm going to try and constrain using black hole super agents zero coupling some matter in mass ranges from 10 to minus 9 to 10 to minus 20 electron volt Okay, so I'm going to only assume you can be More complicated, but I'm only going to assume the following dark matter looks something like either a massive Scaler a massive vector a massive tensor Whatever you want. Okay, it has to be a boson. It has to have some mass term, which is sufficiently small Why Why am I doing this? Well because the idea is to use the black hole bomb again. Okay, I don't need a mirror Because the mass term of the scalar field actually, this is obvious from this equation Okay, this a tilde is the separation constant plus mu squared R squared if you look at the synthetic solutions Oh dissolution by the way is for massless fields if you do dissolution for massive fields. This Omega is replaced by this Okay for massive fields So you see that if the if the field has a sufficiently low energy it can't propagate The square root picks up an eye Right, and you either that get exponential damping or you get exponential growth and that's irregular So in practice the mass term works as this mirror Okay, so you thought you give a kick in a scalar field close to a black hole He gets amplified in the original region tries to escape, but he can't because it's massive. It needs to fall back Well, it's going to get amplified there. So you get an instability I'm not working out the details of the instability, but you can it's a very easy exercise to solve this Equation when Omega is sufficiently small you solve it close to the horizon close to infinity You do much asymptotic expansions and you get the result the result is this one. Okay, the time scale so to summarize Massive scalars around the spinning black hole Trigger and instability and the time scale is this one You could think this process has to be completely negligible. It's a tiny field It's a supermassive guy. How are we even discussing this, right? Well, we're discussing this because the time scale for a supermassive black hole and the field of 10 to minus 16 electron Volt is a hundred seconds. I mean, can you believe this? It's a hundred seconds, right? The reason of course So so it has a strong dependence on the mass, right? So as soon as you get away from an ideal mass, you see there's a power 9 here So the timescale grows immensely if you if you make the mass smaller, okay And actually if the tuning to the ideal timescale Happens when the Compton wavelength of this field is roughly of the order of the sparsial radius of your buckle Okay, so then you have the largest coupling between the field and and the buckle So it's a hundred seconds. It's going to do stuff to our universe. Okay, it's going to do things If you actually try to simulate this this was done by my student healthy V tech a few years back For a vector field. It looks something like this This is an almost maximally spinning black hole and we threw in a Vector field with some master. I don't remember which one. Okay. This is how it looks like. It's a dipole mode So the the stuff that grows outside the blog or has to be non axisymmetric Okay, always if you think about the earth moon system If the moon was a ring The effect would not exist, right? You need some non-symmetry. You need dissipation and so on right So whatever you grow here has to be non axisymmetric So the instability gives rise to a non axisymmetric mode. This is an L1M1 mode. Okay The final state we didn't evolve past the final state. This is a real vector field So Maxwell except that we added a small mass term Okay Will and this did something Slightly different because they wanted to actually understand the final state They kind of cheated a little bit So they added they considered the black hole and now a complex vector field. So it's not really universe It's complex But the advantage of doing a complex field is that the stress tensor can be completely stationary Okay, even if the field oscillates the stress tensor can combine in a way that it's truly stationary So what they get is something like this? they start with a With a cloud around the black hole of very small amplitude the black hole is spinning it doesn't look like it The cloud is extracting angular momentum. It doesn't look like it because it's a complex vector field So the stress tensor again is time independent and in fact, it's very glissimetric Okay, and the cloud just grows. So the final state is a truly hairy black hole solution Okay stationary black hole solution. It's a spinning black hole Surrounded by a co-rotating cloud of vector fields Okay But now back to our universe back to real fields Generating non-oxysymmetric clouds. So question What is going to happen? How is the system going to evolve? Can we use the evolution to do some science or not? So for that you need to evolve the system. Okay, so take a spinning black hole Take a cloud that's growing by a super radiance take an accretion disk That's releasing stuff into the black hole and just evolve this okay This is one example of what can happen you start with a black hole That's 10 to the force over masses. Okay, and the field you admit the existence of a field Which is 10 to the minus 18 electron volt Very light field. Okay, this ability is really weak Okay, it's weak because the mass coupling is only 10 to the minus 4 Okay, very small super radiance is irrelevant, but there's accretion So you start with a black hole. So this is spinning at half the maximum possible value Accretion means that the mass so this is time and I think this is oh, this is Dimensional spin. Okay, so as time goes by the mass increases the guy is eating stuff It's eating stuff that's co-rotating. So the spin increases the spin becomes nearly maximum The mass is still increasing which means that this coupling is also increasing When the coupling become becomes of order one the time scales become of order a minute Okay, then the instability kicks in the black hole loses mass almost instantaneously. Okay, it loses spin and Then super radiance controls everything. Okay, it's still a greeting, but super radiance to cold So the evolution is governed by accretion locked onto super radiance and then you see the black hole follows this line here Okay, in the right you plane black hole mass versus angular momentum After sometime the black hole does this. Okay, of course if you start with a black hole, which is initially more massive Then super radiance kicks in earlier because the coupling is larger. That's it This is one black hole if you spread black holes in your universe You can follow the trajectories. Okay, and you're going to find something that I know you already thought of which is There's some range is some part of barometer space where black holes can't exist Because they would be unstable. They lose mass and spin very quickly. Okay, you see we spread Randomly tend to the cube black holes random spin and mass distributions you evolve them after a Hubble timescale and you find this This is for different Accretion efficiencies, okay After 10 to the 80 years or 10 to the 9 If you look at your black holes in the universe, you will find there shouldn't be any in some parts of barometer space Okay, so now you can think okay So the only thing I need to do is observe black holes measure their mass their spin and If they happen to lie here, this means the field cannot exist. This is the idea, right? Because I mean they would just be unstable there. So if you play this game for vectors You get something like this window here. Okay, so for instance if you admit that there's a vector field of mass 10 to minus 18 In this blue curve here Here all black holes in the universe should be unstable on timescales of a thousand years or less You should not be seeing any Okay, and yet a catalog of black holes. These are these crosses here the size of the cross means observational error Tells you they exist. So these guys ruled out Okay, if you trust the measurements and now you also understand that the way to constrain Lighter and lighter fields is by picking up more and more massive black holes The more massive The massiver black hole we knew at the time was this guy here called feral line If you talk to the persons who did the study, they will tell you they don't trust the measurement errors Okay, but if we trust them The black hole is a mass of roughly 10 to the 9 solar masses This means these guys will be excluding a vector field with a mass 10 to minus 20 electron volt If you look in the particle data group booklet, you will see that accelerator bounds are 10 to minus 18 Okay, so you can use black holes to improve on the CERN physics kind amazing. Okay, of course I'm sweeping under the four Things things meaning these Evolutions never take into account the coupling of the vector field for instance with the accretion discs, right? I don't know if this vector field is interacting a lot with the accretion disc I know if that if I have a massive gravity run the coupling is negligible. So for gravity turns in fact This bound is now on the particle data booklet 10 to the minus 23 electron volts just from super radians of massive vectors Let me quit do I have five minutes? Okay, do I still have five minutes? Oh, okay, then I have plenty of time. Great But I thought I was really running out. Well, that's fantastic So these are measurements using electromagnetic data, okay But the cloud is non-axis symmetric. So you're generating stuff That's spinning Corrotating with a black hole, but it's not axis symmetric. What this does is generate a non-zero time-varying quadruple moment And if you go at Stas Lectures, you will see that a non-zero quadruple moment that varies in time has to emit gravitational waves Okay, so what you measure are these guys. So you measure Black hole spin j over m squared and mass. How do you measure this? Okay, is that is that the question? Okay mass is easy to measure you just look at motion of things nearby Spin is a lot difficult a lot more difficult. I think that's why I Was in Paris giving a talk last week and the author of this guy of this point here don't we don't trust that number The way you measure spin you measure it in two different ways the first is So you I told you that black holes are surrounded by a Christian discs and That the a Christian disc Cuts off at some point called which I call these co the last point where circular orbits are possible Right. So if you look at the Christian discs, you should always see and Radiation from the Christian discs You should always see a maximum frequency in the problem and corresponds to these go now these The location of these go Depends on the buckle spin if you spin this guy up these co is moving inwards. Okay, so one of the ways to do this is to measure Brightness distribution across the a Christian disc and try to match it to some modeling that you have for curve locals in gr Okay, this will be the continuum feeding method. Okay. It's not entirely trivial. There's a lot of parameters going in there And I'm not going to say I don't trust it, but it's I mean it's too complicated for me. Okay the second method Actually seems nicer because you only look at spectral lines. Okay, you look at iron K alpha line It's well, it's one of the spectral lines and an interesting feature of those lines which basically Amounts to what I said about the way that black holes look like is that spectral lines Go get distorted Depending on where you measure them they get Doppler shifts because the atoms are moving they get Gravitational redshift because they're in a potential well and so on right so spectral lines a line That would be looking like this can look like so if you add spin to it can look something like this, right? So people fit the spectral lines in a Christian discs To what you expect from gr So you have two independent completely independent measurements Most of the times they give numbers Which don't agree at all? Okay, there are I think now there's an effort to bring these guys together I think they started getting measurements that agree. It's still a tricky business, but this is how they do it Okay, it can only improve in the future at least Okay But so this cloud is non axisymmetric it's emitting gravitational waves So one of the ways that you have of testing for the presence of fields in Around wackles is to actually just go to LIGO and see if there's Non-trivial sources of waves there, right now the cloud This cloud that develops around the spinning buckle is at most 10% of the buckle mass, okay? You cannot use the quadruple formula to estimate gravitation radiation because the fields are incoherent So that's something we learned when we were trying to do this So the way to actually compute gravitation radiation is exactly the way that we computed Gravitational radiation from point particles here or from scalar fields and so on right you expand the Einstein field equations But the right-hand side corresponds to the stress tensor of this cloud. This is how it's done Okay But you will find that the emission is basically monochromatic Okay, it this system emits waves at a frequency which is roughly Twice the orbital frequency the angular frequency of the buckle which is the orbital frequency of the cloud right, so as Emina, Arvani, Taki and other people Did this very interesting study they looked at At the waves that would come out of the system and they estimated how much number of the number of events that Leica would see per year And this is number if there is because so let me wait. Let me go back There's an important point here if there is a field Somewhere if there's a field a fundamental field with a mass 10 to the minus 12 you like to involve for example Okay, you don't need to have the field present close to the buckle Okay, you don't need to have any abundance of the field anywhere in the universe any small quantum fluctuation of that field is going to grow Exponentially anything is going to trigger the instability. Okay, this means that any Any buckle in the universe eventually would be unstable against this mechanism You don't need to be finding another buckle to merge it with you just need to have a single spinning buckle Eventually it will get there, right So you should be seeing a lot of stuff and in fact Their estimate is up to 10 to the 5 events per year in LIGO If there is a mass if there is a field a scalar with a mass of 10 to minus 13 or so Okay, just because there's so many stellar mass buckles out there For Lisa so you need a lot of more massive buckles with Actually the distribution is kind of uncertain But they estimate that we would see so these these are individual Resolvable events from each of these guys. Okay, they estimate they would see around up to one event per year with Lisa Now we're talking about masses of 10 to minus 17 electron volt Coming from 10 to the 6 or mass buckles 10 to the 8 10 to the 9 But if you think that any buckle in the universe history was subjected to this kind of mechanism You could think oh wait a minute. The universe should be completely filled with radiation from these guys So you could lose you could look for stochastic backgrounds cross-correlating different arms of your of your detector So we did this exercise a few months back and what we found was this So this would be so this is the the density in gravitational waves the primordial density Across different frequencies. This black line is what LIGO is actually what Lisa is supposed to be Measuring when it goes up. Okay Here it's the same for LIGO. This is the first run This will be advanced LIGO. Okay If a field with mass 10 to the minus 17 electron volt exists it would generate this density in waves Okay, clearly clearly above the Lisa threshold So it should be filling the universe with waves of frequency 10 to minus 3 Hertz and we should be seeing them in detectors Okay if a field of a mass 10 to the minus 12 electron volt exists It can't exist because even the first run of LIGO ruled it out. It's a very small window. I mean, sorry This is a very small constraint, okay But still it ruled out exactly 10 to minus 12 10 to minus 12 point one But the the other runs are going to constrain a larger window right so this fantastic We look for stochastic backgrounds in frequency of 10 to 100 Hertz if we don't find it You're really ruling out Fundamental fields that might exist scalars vectors dancers, whatever You could if the curve is similar enough you could confuse it with others since we haven't seen anything yet That these are ruled out anyway, but yeah, there's a danger. This is a very specific profile, right? It's monochromatic. It's a the power law of density. We know it very well It's different from white dwarf white dwarf. So it's different from most of the sources. We know of so And of course now you can ask what about stars If Michael's do this wonderful job can stars do the same and the answer is I have no idea Well, you need couplings. You always need coupling. You remember the earth moon, right? You need the ocean to have some kind of interaction with the crust So if you throw say an electromagnetic wave towards a rotating star if the star doesn't see The the field you cannot I mean, there's nothing there. There's no interaction, right? So you need to do some coupling, okay? Back on sign 10 or well, I don't know 15 years ago in 93. I think 98 my god in 98 Actually showed the following if you have So it's so he was trying to basically Tell people to go to the lab and show that rotational super radiance exists Okay, so he showed the following if you have a spinning Cylinder made of some conducting material with some conductivity Absolute, okay If you throw in so you spin the cylinder it's in this room you spin the cylinder It's spinning with an angular frequency capital omega if you send in waves away from this lamp Not exactly something has to be low frequency But if the frequency of the wave that you send in light is smaller than this Then super radiance occurs. You're going to extract energy away from the rotating cylinder Okay, of course he does this because there's some conductivity of the symbol usually sigma Okay, because there's conductive. There's an interaction between the cylinder material and the light, okay? So it's a very nice Kind of setup. Nobody went to the lab as far as I know to try to see this even Told people look if you surround if you close this in a box, you're going to get an instability So that might be easier to detect. Nobody went to the lab What they did eventually last year in Ottingham was to go to lab with water They made water spin up and it's sauce operators in that way with sound waves But anyway, this tells you that if you are able to couple a star With light then you're going to get super radiance. You can do this. I mean stars have some conductivity Neutron stars for instance have a conductivity. It's really Large, okay, and that's unfortunate They conduct very well But still if you work out the details, which are not very well worked out is even though it's my work We can't control rapidly spinning stars. Okay But you will find that this mechanism does exactly the same if you have a massive field Around the star that has a conductivity for that kind of field The field interacts with the star extracts a bit of energy. It's massive. So it can propagate goes back Extract a bit of energy, right? So stars are going to spin down the wonderful thing about neutron stars is that some of them are pulsars and Pulsars the period of pulsars are on to a very very very very very good precision So any spin down you're going to see it, right? So you can impose constraints there. They are what they are. I don't think they are Ultra interesting. I hope nobody's taping this but still if you plot The conductivity versus mass plane These are the constraints that you can put that you can put for different pulsars. These are here Okay, so in this range of conductivity versus mass These guys would spin down in a way that you would see them and we don't see them spinning down. Okay to that rate So it's kind of interesting There's other things that I slept I always like to give the stuff that we don't know and I think unfortunately we don't know most of the things Okay, so these fields Mostly these scalars that I've been talking they were introduced mostly as pseudo scalars Right, and there should be some some coupling to standard model fields. I've been neglecting them Okay, if you don't neglect them one Possible coupling goes like this, right? So you have the field you have the master Superavians here in stability here cloud here. That's good, but what happens if you look at this guy Well, now you have a field that's growing But who knows If it's really growing because it's talking to a massless field So the energy that it extracts from the black hole could simply be transferred via this coupling to To Maxwell and Maxwell goes away. That's it. No cloud. No nothing, right? For most of the copings we know of these guys are small enough that this effect is small But a few months ago actually it came out last week on PRL These guys here conjectured that well, they have some calculation not very Well, not yet all the things that you would expect but they conjecture that what's going to happen When the coupling is taken into account is the following These guys make the cloud grow so spinning black hole cloud growing growing growing Then at some point this guy because you see what matters is k-axis times the magnitude of the scalar This guy becomes large enough and you're basically going to trigger a blast of electromagnetic waves suddenly This goes over a critical threshold and you get an explosion in Electromagnetic waves they call these blasts. I don't know the acronym what it stands for. But anyway, it looks nice They also conjecture that it only happens for black hole masses smaller than 10 to the minus 2 solar masses So this can only apply for primordial black holes, but we need Urgently a real calculation of this. Okay, they have estimates, but that's really all there is to it I think I'm going to skip this The other open questions. So these are open questions that I had 1.5 years ago Some of them I think are being resolved, but I think many of them will stay with us for years The first is can we measure rotational super events? I was giving this talk in Nottingham I think 1.5. Well, maybe more years ago and fortunately for me those guys close the subject We did measure it in the lab on earth That's good. If you work out super radians, it comes in all bunches of forms Charged black holes have super radians against charged scalar fields spinning Water that's spinning your bathtub has super radians against sound waves. Okay I think below who conjectures that cyclones also have super radians playing your role there If you work out the maximum amplification factor for any of the systems we know Something funny seems to happen Okay For massless fields the amplification factor is at most a hundred and forty percent For charged fields, it's exactly bounded by a hundred percent For acoustic black holes, it's ex bounded by exactly a hundred percent. Okay And for every case that I looked at I could never get amplification factors So I send in a wave of one. I never get more than two back. Okay Maybe this is just a coincidence or maybe there's something deeper at work that I have no idea where it is But I thought it would be nice to live this year. So it's an open issue for me Another open issue very important that we need to understand. I think urgently in the next decades are black hole binaries We know they're a single black hole has ergo regions We have we don't even know how to define an ergo region when we have dynamical space times where this lousy, right? Of course, you could take a numerical relativity simulation Throwing grenades let them explode and measure the outgoing partial. This has not been done But I think somebody should take one year of their life to do this because it's really interesting Do black hole binaries give rise to super radians or not? We have no idea I mean the numerical relativity simulations you find out there even the best ones Evolve up to 20 50 the best one a hundred orbits. This is nothing, right? The universe will produce a hundred thousands millions of orbits that we're going to see in the detectors The timescales are completely different. We have no idea of what happens on these huge timescales The end state of super-radial disability somebody asked The end state of course is always a slower spinning black hole in a syntotically flat Space times if your field is real just because Gravitational ex-exists they give rise to the no-hair theorems, okay? So they they basically work to spin down everything we know the universe really hates relative motion In ABS in in space times, which are slightly different. We don't I don't think we know, okay? Still nobody evolved fully numerically a Super radiant in stability in flat space time taking a simulation putting a black hole there Triggering a scalar field or vector field letting it go see the cloud grow see gravitation with the mission and so on all of this has been done At perturbative level using exactly the tools we discussed in these four lectures Okay, this is all there is we think it's accurate enough, but who knows? It's usually stated that super radiance is the exact wave analog of the Penrose process But the Penrose process was done if you remember with grenades we throw a particle in it separates and that's it But these guys are made of fermions Okay fermions have no super radiance just because of Paulie exclusion principle You can't really have two guys occupying the same state. So if you throw in a Fermion into a spinning black hole, you're not going to get energy extraction. Okay, so one of the puzzles is how come the Penrose process works for fermions and It's supposed to be the analog of the super radiance which only works for for Rosens. How can he you know? How can you kind of make sense of this? There was some partial progress by a student of mine, but I don't think this is nearly finished really and Finally, this is a key question. It will maybe take years. Maybe it takes weeks. I have no idea We need a great idea. We need somebody to sit down and work out what happens to a super radiance when you have real universe When you have a Christian this when you have magnetic fields and it couples to your to your boson Finally before finishing Let me tell you another thing that happens. So this is this is an example that I did to show you what's super radiance By the way super is was known by Klein in the 20s 1920s. Okay This was done. He was trying to investigate what happens to fermions when you Scatter them off a potential barrier like this one. This can be our effective potential in bugle space times, okay? If you scatter a charged field, so this is equation of motion Okay, you will find that exactly the same thing happens. You throw in a wave But what the way I bounce back so in a wave a fraction goes here. He has negative energy So this guy was amplified at the expense of a negative energy inside the barrier Okay, exactly the same thing happens for a buckle the potential looks slightly different. It's not constant You're exactly the same. Okay, and you can see that because there is an horizon here This way the negative energy state just keeps keeps on going. So we don't care about it But now you can think ah, wait a minute Let me try to connect to what he was saying in the morning if there is no horizon, right? I told you already that this guy has to bounce back. So this is what happens. I throw in the wave packet But now there's something here. So it has to come back It interacts with the with the border and you see it creates a new a new pulse creates a new pulse The negative energy state is growing Therefore this pulse is also growing and you see what you're doing, right? You're basically just triggering an instability because there's no horizon to dump the negative energy states into So very nice You can actually use this and a lot of ignorance to to constrain Potential objects which do not have horizons. So if you admit it's a huge assumption Okay, but if you admit that the compact objects out there have an exterior metric that's curve But cut off at some radius Okay, you can work out the details of this instability and again if all of the object in the universe If none of the objects in the universe are vehicles. This is going to give you a huge stochastic background Actually, let me see. Yeah, this is stochastic background. It's huge. So if Basically the non measurement of any stochastic background in gravitation of detectors rules out all of these models Anything that looks like current the outside, but does not have an horizon would be ruled out. Okay. This is an example Where let me see what I have. Oh, yeah, so I'm placing I'm taking a current geometry and I'm cutting it off at the radius R Which is the horizon one plus epsilon and epsilon is 10 to minus 40 Okay Right If so, this is a conservative number by the way effects on I will explain why in a minute 10 to minus 40 This line here will basically you see this will be the level of the background that you'd get Advanced logo design would see all of this all of this So no object even with an epsilon of 10 to minus 40 could exist in the universe under this assumption. Okay If you make a epsilon Even smaller you could think oh wait a minute. I can go around this if actually is really really small You look at the way the way go. So let me wait Look at this if epsilon is really small. It means that this wall is going to the left So if he really goes to the left Then the wave takes a lot of time to get there and the time the instability timescale will become larger So you think okay, I can get around this if a epsilon is really small. Okay The problem with that is that the timescale the light if you follow light rays We did this exercise in the first class the timescale the time that light takes to hit a barrier Goes like log of epsilon Okay, so even if you make epsilon 10 to minus 100, which is a minute number It's going to change this by a factor 2, right? So it doesn't really really help you Of course, you can also argue. Well, wait a minute if there's viscosity somewhere in the problem that might kill The instability that's true I don't think we know anything about absorption and so I'm not even going to mention that so in the Assumption that the exterior metric is current And nothing else happens There's no funny business going on inside the star then you rule out basically everything by the non-observation of gravitation wave backgrounds so let me finish this talk and my set of lectures by Trying to convey the message that really these are exciting times for me But mostly for you for people who are starting the PhD or young postdocs. These are really unique times I started my PhD as a crackpot Somebody going around looking at gravitational waves giving talks about what the guy is saying again And we've been looking for waves for 20 years 50 years. They don't exist. We'll never see them These detectors cannot see stuff that moves on 10 to minus 21 scales, right? So I think you're in a much better position that at least we saw them It's the time to understand what they're telling us and we have no idea. I mean this You know, I think sometimes I sound way more excited than I should be But that's these are just a couple of things that we thought about in the last two years There's really a lot of stuff to be found out a lot of stuff to be done So thank you and enjoy the meeting